In "The Scarlet Letter," why does Hester speak to Roger Chillingworth after so many years? What does she hope to accomplish?

In one sense she didn't have a choice.  It was her and Pearl's emotional turmoil that led them to seek help for her.  As Chapter 4 describes it when Chillingworth is brought in: 

"I shall own you [Chillingworth] for a man of skill indeed! Verily, the woman hath been like a possessed one; and there lacks little, that I should take in hand to drive Satan out of her with stripes."  

Although Hester is fearful and suspects that Chillingworth might try to poison her or Pearl, she meets a man quite different from what she had expected.  "Foolish woman!" responded the physician, half coldly, half soothingly. "What should ail me to harm this misbegotten and miserable babe?"

Thus begins their long discussion about Chillingworth's desire (soon to be come obsession) to find out who the father is.   But before that, there is the somewhat touching conversation where they both confess to contributing to the failure of their marriage.  This is also the point at which Hester, much to her later regret, promises to keep his identity secret.

 It is interesting how Hawthorne brought them both together and allowed us to understand all there is to know about their past relationship while setting the stage for what is to come.